Arithmetic properties of direct product of $p$-adic fields elements

The article considers the transcendence and algebraic independence problems, introduce statements and proofs of theorems for some kinds of elements from direct product of $p$-adic fields and polynomial estimation theorem. Let $\mathbb{Q}_p$ be the $p$-adic completion of $\mathbb{Q}$, $\Omega_{p}$ be the completion of the algebraic closure of $\mathbb{Q}_p$, $g=p_1p_2\ldots p_n$ be a composition of separate prime numbers, $\mathbb{Q}_g$ be the $g$-adic completion of $\mathbb{Q}$, in other words $\mathbb{Q}_{p_1}\oplus\ldots\oplus\mathbb{Q}_{p_n}$. The ring $\Omega_g\cong\Omega_{p_1}\oplus\ldots\oplus\Omega_{p_n}$, contains a subring $\mathbb{Q}_g$. The transcendence and algebraic independence over $\mathbb{Q}_g$ are under consideration. Here are appropriate theorems for numbers like $\alpha=\sum\limits_{j=0}^{\infty}a_{j}g^{r_{j}}$, where $a_{j}\in \mathbb Z_g,$ and non-negative rational numbers $r_{j}$ increase to strictly unbounded.